A theorem of lattice theory states that any
non-distributive non-modular lattice contains N5 as a sublattice.

Now consider this Brownian form; “two ducks in a box”:

C=[[a[a]]a[b[b]]bc]c

_________________________________

______________________|a = [a[a]]

|_____|_____||

|| | || | ||b = [b[b]]

|| | | || | | ||

||___|_||___|_||c = [abc]

|_____________________________|

In the nand interpretation, this is:

a=Da=“I am honest or a liar.”

b=Db=“I am honest or a liar.”

c=~ (a &
b & c)=“One of us is a liar.”

Note that sentence c is of the form

c=c -> ( da V
db )

- which is Boolean only if the lower differentials disjoin
to true.

It has this fixedpoint lattice:

iti -------- ijt -------- tjj

/\/\

/\/\

iiitti ----------- ttjjjj

\/\/

\/\/

tii -------- jit -------- jtj

In the nor interpretation, this is:

a=da=“I am honest and a liar.”

b=db=“I am honest and a liar.”

c=~ ( a V
b V c )=“All
of us are liars.”

Note that sentence c is of the form:

c=( Db & Db ) - c

- which is Boolean only if the upper differentials conjoin
to false.

It has this fixedpoint lattice:

ifi --------- ijf --------- fjj

/\/\

/\/\

/\/\

iiiffi ----------- ffjjjj

\/\/

\/\/

\/\/

fii --------- jif --------- jfj

Note the fixedpoints ijf and jif; these are the only
one where C has a boolean value; but this is due to Complementarity, an
anti-boolean axiom. Without those points, this lattice would be modular and
distributive; but with them it contains N5.